def run_optimization(sim_res, time, price, pv, bldg, plot = True, usePV = True):
"""
This function runs an optimization problem
"""
from pyjmi.optimization.casadi_collocation import MeasurementData
from collections import OrderedDict
# get current directory
curr_dir = os.path.dirname(os.path.abspath(__file__));
# if pv panels are not used then remove power
if usePV == False:
pv = np.zeros(np.shape(pv))
# compile FMU
path = os.path.join(curr_dir,"..","Models","ElectricalNetwork.mop")
model_name = compile_fmux('ElectricNetwork.ACnetworkBatteryMngmtOpt_Money', path, compiler_options={"enable_variable_scaling":True})
# Load the model
model_casadi = CasadiModel(model_name)
# Get the inputs that should be eliminated from the optimization variables
eliminated = OrderedDict()
data_price = np.vstack([t_data, price])
eliminated['price'] = data_price
#data_Q = np.vstack([t_data, np.zeros(Npoints)])
#eliminated['Q_batt'] = data_Q
data_pv1 = np.vstack([t_data, np.squeeze(pv[:,0])])
eliminated['P_pv1'] = data_pv1
data_pv2 = np.vstack([t_data, np.squeeze(pv[:,1])])
eliminated['P_pv2'] = data_pv2
data_pv3 = np.vstack([t_data, np.squeeze(pv[:,2])])
eliminated['P_pv3'] = data_pv3
data_bldg1 = np.vstack([t_data, -np.squeeze(bldg[:,0])])
eliminated['P_bldg1'] = data_bldg1
data_bldg2 = np.vstack([t_data, -np.squeeze(bldg[:,1])])
eliminated['P_bldg2'] = data_bldg2
data_bldg3 = np.vstack([t_data, -np.squeeze(bldg[:,2])])
eliminated['P_bldg3'] = data_bldg3
measurement_data = MeasurementData(eliminated = eliminated)
# define the optimization problem
opts = model_casadi.optimize_options()
opts['n_e'] = 60
opts['measurement_data'] = measurement_data
opts['init_traj'] = sim_res.result_data
# get the results of the optimization
res = model_casadi.optimize(options = opts)
if plot:
plotFunction(sim_res, res)
return res
def run_optimization(sim_res, time, price, pv, bldg):
"""
This function runs an optimization problem
"""
from pyjmi.optimization.casadi_collocation import MeasurementData
from collections import OrderedDict
# get current directory
curr_dir = os.path.dirname(os.path.abspath(__file__))
# compile FMU
path = os.path.join(curr_dir, "..", "Models", "ElectricalNetwork.mop")
model_name = compile_fmux(
'ElectricNetwork.NetworkBatteryMngmtOpt_Money',
path,
compiler_options={"enable_variable_scaling": True})
# Load the model
model_casadi = CasadiModel(model_name)
# Get the inputs that should be eliminated from the optimization variables
eliminated = OrderedDict()
data_price = np.vstack([t_data, price])
eliminated['price'] = data_price
data_pv1 = np.vstack([t_data, np.squeeze(pv[:, 0])])
eliminated['pv1'] = data_pv1
data_pv2 = np.vstack([t_data, np.squeeze(pv[:, 1])])
eliminated['pv2'] = data_pv2
data_pv3 = np.vstack([t_data, np.squeeze(pv[:, 2])])
eliminated['pv3'] = data_pv3
data_bldg1 = np.vstack([t_data, np.squeeze(bldg[:, 0])])
eliminated['bldg1'] = data_bldg1
data_bldg2 = np.vstack([t_data, np.squeeze(bldg[:, 1])])
eliminated['bldg2'] = data_bldg2
data_bldg3 = np.vstack([t_data, np.squeeze(bldg[:, 2])])
eliminated['bldg3'] = data_bldg3
measurement_data = MeasurementData(eliminated=eliminated)
# define the optimization problem
opts = model_casadi.optimize_options()
opts['n_e'] = 50
opts['measurement_data'] = measurement_data
opts['init_traj'] = sim_res.result_data
# get the results of the optimization
res = model_casadi.optimize(options=opts)
plotCompare(sim_res, res)
def run_optimization(sim_res, time, price, pv, bldg):
"""
This function runs an optimization problem
"""
from pyjmi.optimization.casadi_collocation import MeasurementData
from collections import OrderedDict
# get current directory
curr_dir = os.path.dirname(os.path.abspath(__file__));
# compile FMU
path = os.path.join(curr_dir,"..","Models","ElectricalNetwork.mop")
model_name = compile_fmux('ElectricNetwork.NetworkBatteryMngmtOpt_Money', path, compiler_options={"enable_variable_scaling":True})
# Load the model
model_casadi = CasadiModel(model_name)
# Get the inputs that should be eliminated from the optimization variables
eliminated = OrderedDict()
data_price = np.vstack([t_data, price])
eliminated['price'] = data_price
data_pv1 = np.vstack([t_data, np.squeeze(pv[:,0])])
eliminated['pv1'] = data_pv1
data_pv2 = np.vstack([t_data, np.squeeze(pv[:,1])])
eliminated['pv2'] = data_pv2
data_pv3 = np.vstack([t_data, np.squeeze(pv[:,2])])
eliminated['pv3'] = data_pv3
data_bldg1 = np.vstack([t_data, np.squeeze(bldg[:,0])])
eliminated['bldg1'] = data_bldg1
data_bldg2 = np.vstack([t_data, np.squeeze(bldg[:,1])])
eliminated['bldg2'] = data_bldg2
data_bldg3 = np.vstack([t_data, np.squeeze(bldg[:,2])])
eliminated['bldg3'] = data_bldg3
measurement_data = MeasurementData(eliminated = eliminated)
# define the optimization problem
opts = model_casadi.optimize_options()
opts['n_e'] = 50
opts['measurement_data'] = measurement_data
opts['init_traj'] = sim_res.result_data
# get the results of the optimization
res = model_casadi.optimize(options = opts)
plotCompare(sim_res, res)
def run_demo(with_plots=True):
"""
Demonstrate how to solve a dynamic optimization problem based on a Van der
Pol oscillator system.
"""
curr_dir = os.path.dirname(os.path.abspath(__file__))
jn = compile_fmux("VDP_pack.VDP_Opt2", curr_dir + "/files/VDP.mop")
model = CasadiModel(jn)
opts = model.optimize_options(algorithm='CasadiPseudoSpectralAlg')
opts['n_cp'] = 50
#opts['IPOPT_options']['hessian_approximation'] = 'limited-memory'
res = model.optimize(algorithm='CasadiPseudoSpectralAlg', options=opts)
# Extract variable profiles
x1 = res['x1']
x2 = res['x2']
u = res['u']
time = res['time']
assert N.abs(res.final('x1') - 8.64330006e-07) < 1e-3
assert N.abs(res.final('x2') + 3.68158852e-07) < 1e-3
if with_plots:
# Plot
plt.figure(1)
plt.clf()
plt.subplot(311)
plt.plot(time, x1, 'x-')
plt.grid()
plt.ylabel('x1')
plt.subplot(312)
plt.plot(time, x2, 'x-')
plt.grid()
plt.ylabel('x2')
plt.subplot(313)
plt.plot(time, u, 'x-')
plt.grid()
plt.ylabel('u')
plt.xlabel('time')
plt.show()
def run_demo(with_plots=True):
"""
Demonstrate how to solve a dynamic optimization problem based on a Van der
Pol oscillator system.
"""
curr_dir = os.path.dirname(os.path.abspath(__file__));
jn = compile_fmux("VDP_pack.VDP_Opt2", curr_dir+"/files/VDP.mop")
model = CasadiModel(jn)
opts = model.optimize_options(algorithm='CasadiPseudoSpectralAlg')
opts['n_cp'] = 50
#opts['IPOPT_options']['hessian_approximation'] = 'limited-memory'
res = model.optimize(algorithm='CasadiPseudoSpectralAlg',options=opts)
# Extract variable profiles
x1 = res['x1']
x2 = res['x2']
u = res['u']
time = res['time']
assert N.abs(res.final('x1') - 8.64330006e-07) < 1e-3
assert N.abs(res.final('x2') + 3.68158852e-07) < 1e-3
if with_plots:
# Plot
plt.figure(1)
plt.clf()
plt.subplot(311)
plt.plot(time,x1,'x-')
plt.grid()
plt.ylabel('x1')
plt.subplot(312)
plt.plot(time,x2,'x-')
plt.grid()
plt.ylabel('x2')
plt.subplot(313)
plt.plot(time,u,'x-')
plt.grid()
plt.ylabel('u')
plt.xlabel('time')
plt.show()
def run_demo(with_plots=True):
"""
Demonstrate how to optimize a VDP oscillator using LocalDAECollocationAlgOld.
"""
# Compile and load model
curr_dir = os.path.dirname(os.path.abspath(__file__));
jn = compile_fmux("VDP_pack.VDP_Opt2", curr_dir + "/files/VDP.mop")
model = CasadiModel(jn)
# Set algorithm options
opts = model.optimize_options()
opts['graph'] = "SX"
# Optimize
res = model.optimize(options=opts)
# Extract variable profiles
x1 = res['x1']
x2 = res['x2']
u = res['u']
time = res['time']
assert N.abs(res.final('x1')) < 1e-3
assert N.abs(res.final('x2')) < 1e-3
# Plot
if with_plots:
plt.figure(1)
plt.clf()
plt.subplot(3, 1, 1)
plt.plot(time, x1)
plt.grid()
plt.ylabel('x1')
plt.subplot(3, 1, 2)
plt.plot(time, x2)
plt.grid()
plt.ylabel('x2')
plt.subplot(3, 1, 3)
plt.plot(time, u)
plt.grid()
plt.ylabel('u')
plt.xlabel('time')
plt.show()
def run_demo(with_plots=True):
curr_dir = os.path.dirname(os.path.abspath(__file__));
jmu_name = compile_jmu("Orbital", curr_dir+"/files/Hohmann.mop")
comp_opts = {"normalize_minimum_time_problems": False} # Disable minimum time normalization
fmux_name = compile_fmux("HohmannTransfer", curr_dir+"/files/Hohmann.mop",
compiler_options=comp_opts)
# Optimization
model = CasadiModel(fmux_name)
opts = model.optimize_options(algorithm="CasadiPseudoSpectralAlg")
opts["n_cp"] = 40 # Number of collocation points
opts["n_e"] = 2 # Number of phases
opts["free_phases"] = True # The phase boundary is allowed to be changed in time
opts["phase_options"] = ["t1"] # The phase boundary is connected to variable t1
opts["link_options"] = [(1,"vy","dy1"),(1,"vx","dx1")] # Allow for discontinuities between phase 1 and 2 for vy and vx.
# The discontinuities are connected by dy1 and dx1
# Optimize
res_opt = model.optimize(algorithm="CasadiPseudoSpectralAlg", options=opts)
# Get results
dx1,dy1,dx2,dy2 = res_opt.final("dx1"), res_opt.final("dy1"), res_opt.final("dx2"), res_opt.final("dy2")
r1,r2,my = res_opt.final("rStart"), res_opt.final("rFinal"), res_opt.final("my")
tf,t1 = res_opt.final("time"), res_opt.final("t1")
# Verify solution using theoretical results
# Assert dv1
assert N.abs( N.sqrt(dx1**2+dy1**2) - N.sqrt(my/r1)*(N.sqrt(2*r2/(r1+r2))-1) ) < 1e-1
#Assert dv2
assert N.abs( N.sqrt(dx2**2+dy2**2) - N.sqrt(my/r2)*(1-N.sqrt(2*r1/(r1+r2))) ) < 1e-1
#Assert transfer time
assert N.abs( tf-t1 - N.pi*((r1+r2)**3/(8*my))**0.5 ) < 1e-1
# Verify solution by simulation
model = JMUModel(jmu_name)
# Retrieve the options
opts = model.simulate_options()
opts["ncp"] = 100
opts["solver"] = "IDA"
opts["initialize"] = False
# Simulation of Phase 1
res = model.simulate(final_time=t1,options=opts)
x_phase_1,y_phase_1 = res["x"], res["y"]
# Simulation of Phase 2
model.set("vx", dx1 + res.final("vx"))
model.set("vy", dy1 + res.final("vy"))
res = model.simulate(start_time=t1,final_time=tf,options=opts)
x_phase_2,y_phase_2 = res["x"], res["y"]
# Simulation of Phase 3 (verify that we are still in orbit)
model.set("vx", dx2 + res.final("vx"))
model.set("vy", dy2 + res.final("vy"))
res = model.simulate(start_time=tf, final_time=tf*2, options=opts)
x_phase_3,y_phase_3 = res["x"], res["y"]
if with_plots:
# Plot Earth
r = 1.0
xE = r*N.cos(N.linspace(0,2*N.pi,200))
yE = r*N.sin(N.linspace(0,2*N.pi,200))
plt.plot(xE,yE,label="Earth")
# Plot Orbits
r = res.final("rStart")
xS = r*N.cos(N.linspace(0,2*N.pi,200))
yS = r*N.sin(N.linspace(0,2*N.pi,200))
plt.plot(xS,yS,label="Low Orbit")
r = res.final("rFinal")
xSE = r*N.cos(N.linspace(0,2*N.pi,200))
ySE = r*N.sin(N.linspace(0,2*N.pi,200))
plt.plot(xSE,ySE,label="High Orbit")
# Plot Satellite trajectory
x_sim=N.hstack((N.hstack((x_phase_1,x_phase_2)),x_phase_3))
y_sim=N.hstack((N.hstack((y_phase_1,y_phase_2)),y_phase_3))
plt.plot(x_sim,y_sim,"-",label="Satellite")
# Plot Rocket Burns
plt.arrow(x_phase_1[-1],y_phase_1[-1],0.5*dx1,0.5*dy1, width=0.01,label="dv1")
plt.arrow(x_phase_2[-1],y_phase_2[-1],0.5*dx2,0.5*dy2, width=0.01,label="dv2")
plt.legend()
plt.show()
def run_demo(with_plots=True):
"""
This example is based on the Hicks-Ray Continuously Stirred Tank Reactors
(CSTR) system. The system has two states, the concentration and the
temperature. The control input to the system is the temperature of the
cooling flow in the reactor jacket. The chemical reaction in the reactor is
exothermic, and also temperature dependent; high temperature results in high
reaction rate.
The problem is solved using the CasADi-based collocation algorithm. The
steps performed correspond to those demonstrated in
example pyjmi.examples.cstr, where the same problem is solved using the
default JMI algorithm. FMI is used for initialization and simulation
purposes.
The following steps are demonstrated in this example:
1. How to solve the initialization problem. The initialization model has
equations specifying that all derivatives should be identically zero,
which implies that a stationary solution is obtained. Two stationary
points, corresponding to different inputs, are computed. We call the
stationary points A and B respectively. Point A corresponds to
operating conditions where the reactor is cold and the reaction rate is
low, whereas point B corresponds to a higher temperature where the
reaction rate is high.
2. How to generate an initial guess for a direct collocation method by
means of simulation with a constant input. The trajectories resulting
from the simulation are used to initialize the variables in the
transcribed NLP.
3. An optimal control problem is solved where the objective is to transfer
the state of the system from stationary point A to point B. The
challenge is to ignite the reactor while avoiding uncontrolled
temperature increase.
4. Finally the system is simulated using the optimal control profile. This
step is important in order to verify that the approximation in the
transcription step is sufficiently accurate.
"""
### 1. Solve the initialization problem
# Locate the Modelica and Optimica code
file_path = os.path.join(get_files_path(), "CSTR.mop")
# Compile the stationary initialization model into a FMU
init_fmu = compile_fmu("CSTR.CSTR_Init", file_path)
# Load the FMU
init_model = load_fmu(init_fmu)
# Set input for Stationary point A
Tc_0_A = 250
init_model.set('Tc', Tc_0_A)
# Solve the initialization problem using FMI
init_model.initialize()
# Store stationary point A
[c_0_A, T_0_A] = init_model.get(['c', 'T'])
# Print some data for stationary point A
print(' *** Stationary point A ***')
print('Tc = %f' % Tc_0_A)
print('c = %f' % c_0_A)
print('T = %f' % T_0_A)
# Set inputs for Stationary point B
init_model = load_fmu(init_fmu)
Tc_0_B = 280
init_model.set('Tc', Tc_0_B)
# Solve the initialization problem using FMI
init_model.initialize()
# Store stationary point B
[c_0_B, T_0_B] = init_model.get(['c', 'T'])
# Print some data for stationary point B
print(' *** Stationary point B ***')
print('Tc = %f' % Tc_0_B)
print('c = %f' % c_0_B)
print('T = %f' % T_0_B)
### 2. Compute initial guess trajectories by means of simulation
# Compile the optimization initialization model
init_sim_fmu = compile_fmu("CSTR.CSTR_Init_Optimization", file_path)
# Load the model
init_sim_model = load_fmu(init_sim_fmu)
# Set initial and reference values
init_sim_model.set('cstr.c_init', c_0_A)
init_sim_model.set('cstr.T_init', T_0_A)
init_sim_model.set('c_ref', c_0_B)
init_sim_model.set('T_ref', T_0_B)
init_sim_model.set('Tc_ref', Tc_0_B)
# Simulate with constant input Tc
init_res = init_sim_model.simulate(start_time=0., final_time=150.)
# Extract variable profiles
c_init_sim = init_res['cstr.c']
T_init_sim = init_res['cstr.T']
Tc_init_sim = init_res['cstr.Tc']
t_init_sim = init_res['time']
# Plot the initial guess trajectories
if with_plots:
plt.close(1)
plt.figure(1)
plt.hold(True)
plt.subplot(3, 1, 1)
plt.plot(t_init_sim, c_init_sim)
plt.grid()
plt.ylabel('Concentration')
plt.title('Initial guess obtained by simulation')
plt.subplot(3, 1, 2)
plt.plot(t_init_sim, T_init_sim)
plt.grid()
plt.ylabel('Temperature')
plt.subplot(3, 1, 3)
plt.plot(t_init_sim, Tc_init_sim)
plt.grid()
plt.ylabel('Cooling temperature')
plt.xlabel('time')
plt.show()
### 3. Solve the optimal control problem
# Compile model
fmux = compile_fmux("CSTR.CSTR_Opt2", file_path)
# Load model
cstr = CasadiModel(fmux)
# Set reference values
cstr.set('Tc_ref', Tc_0_B)
cstr.set('c_ref', c_0_B)
cstr.set('T_ref', T_0_B)
# Set initial values
cstr.set('cstr.c_init', c_0_A)
cstr.set('cstr.T_init', T_0_A)
# Set options
opt_opts = cstr.optimize_options()
opt_opts['n_e'] = 100 # Number of elements
opt_opts['init_traj'] = init_res.result_data
# Solve the optimal control problem
res = cstr.optimize(options=opt_opts)
# Extract variable profiles
c_res = res['cstr.c']
T_res = res['cstr.T']
Tc_res = res['cstr.Tc']
time_res = res['time']
c_ref = res['c_ref']
T_ref = res['T_ref']
Tc_ref = res['Tc_ref']
# Verify solution for testing purposes
try:
import casadi
except:
pass
else:
cost = float(res.solver.solver.output(casadi.NLP_SOLVER_F))
assert (N.abs(cost / 1.e3 - 1.8585429) < 1e-3)
# Plot the results
if with_plots:
plt.close(2)
plt.figure(2)
plt.hold(True)
plt.subplot(3, 1, 1)
plt.plot(time_res, c_res)
plt.plot([time_res[0], time_res[-1]], [c_ref, c_ref], '--')
plt.grid()
plt.ylabel('Concentration')
plt.title('Optimized trajectories')
plt.subplot(312)
plt.plot(time_res, T_res)
plt.plot([time_res[0], time_res[-1]], [T_ref, T_ref], '--')
plt.grid()
plt.ylabel('Temperature')
plt.subplot(313)
plt.plot(time_res, Tc_res)
plt.plot([time_res[0], time_res[-1]], [Tc_ref, Tc_ref], '--')
plt.grid()
plt.ylabel('Cooling temperature')
plt.xlabel('time')
plt.show()
### 4. Simulate to verify the optimal solution
# Compile model
sim_fmu = compile_fmu("CSTR.CSTR", file_path)
# Load model
sim_model = load_fmu(sim_fmu)
# Get optimized input
(_, opt_input) = res.solver.get_opt_input()
# Set initial values
sim_model.set('c_init', c_0_A)
sim_model.set('T_init', T_0_A)
# Simulate using optimized input
res = sim_model.simulate(start_time=0.,
final_time=150.,
input=('Tc', opt_input))
# Extract variable profiles
c_sim = res['c']
T_sim = res['T']
Tc_sim = res['Tc']
time_sim = res['time']
# Plot the results
if with_plots:
plt.close(3)
plt.figure(3)
plt.hold(True)
plt.subplot(3, 1, 1)
plt.plot(time_res, c_res, '--', lw=5)
plt.plot(time_sim, c_sim, lw=2)
plt.legend(('optimized', 'simulated'))
plt.grid(True)
plt.ylabel('Concentration')
plt.title('Verification')
plt.subplot(3, 1, 2)
plt.plot(time_res, T_res, '--', lw=5)
plt.plot(time_sim, T_sim, lw=2)
plt.grid(True)
plt.ylabel('Temperature')
plt.subplot(3, 1, 3)
plt.plot(time_res, Tc_res, '--', lw=5)
plt.plot(time_sim, Tc_sim, lw=2)
plt.grid(True)
plt.ylabel('Cooling temperature')
plt.xlabel('time')
plt.show()
def run_demo(with_plots=True):
curr_dir = os.path.dirname(os.path.abspath(__file__))
jmu_name = compile_jmu("Orbital", curr_dir + "/files/Hohmann.mop")
comp_opts = {
"normalize_minimum_time_problems": False
} # Disable minimum time normalization
fmux_name = compile_fmux("HohmannTransfer",
curr_dir + "/files/Hohmann.mop",
compiler_options=comp_opts)
# Optimization
model = CasadiModel(fmux_name)
opts = model.optimize_options(algorithm="CasadiPseudoSpectralAlg")
opts["n_cp"] = 40 # Number of collocation points
opts["n_e"] = 2 # Number of phases
opts[
"free_phases"] = True # The phase boundary is allowed to be changed in time
opts["phase_options"] = [
"t1"
] # The phase boundary is connected to variable t1
opts["link_options"] = [
(1, "vy", "dy1"), (1, "vx", "dx1")
] # Allow for discontinuities between phase 1 and 2 for vy and vx.
# The discontinuities are connected by dy1 and dx1
# Optimize
res_opt = model.optimize(algorithm="CasadiPseudoSpectralAlg", options=opts)
# Get results
dx1, dy1, dx2, dy2 = res_opt.final("dx1"), res_opt.final(
"dy1"), res_opt.final("dx2"), res_opt.final("dy2")
r1, r2, my = res_opt.final("rStart"), res_opt.final(
"rFinal"), res_opt.final("my")
tf, t1 = res_opt.final("time"), res_opt.final("t1")
# Verify solution using theoretical results
# Assert dv1
assert N.abs(
N.sqrt(dx1**2 + dy1**2) - N.sqrt(my / r1) *
(N.sqrt(2 * r2 / (r1 + r2)) - 1)) < 1e-1
#Assert dv2
assert N.abs(
N.sqrt(dx2**2 + dy2**2) - N.sqrt(my / r2) *
(1 - N.sqrt(2 * r1 / (r1 + r2)))) < 1e-1
#Assert transfer time
assert N.abs(tf - t1 - N.pi * ((r1 + r2)**3 / (8 * my))**0.5) < 1e-1
# Verify solution by simulation
model = JMUModel(jmu_name)
# Retrieve the options
opts = model.simulate_options()
opts["ncp"] = 100
opts["solver"] = "IDA"
opts["initialize"] = False
# Simulation of Phase 1
res = model.simulate(final_time=t1, options=opts)
x_phase_1, y_phase_1 = res["x"], res["y"]
# Simulation of Phase 2
model.set("vx", dx1 + res.final("vx"))
model.set("vy", dy1 + res.final("vy"))
res = model.simulate(start_time=t1, final_time=tf, options=opts)
x_phase_2, y_phase_2 = res["x"], res["y"]
# Simulation of Phase 3 (verify that we are still in orbit)
model.set("vx", dx2 + res.final("vx"))
model.set("vy", dy2 + res.final("vy"))
res = model.simulate(start_time=tf, final_time=tf * 2, options=opts)
x_phase_3, y_phase_3 = res["x"], res["y"]
if with_plots:
# Plot Earth
r = 1.0
xE = r * N.cos(N.linspace(0, 2 * N.pi, 200))
yE = r * N.sin(N.linspace(0, 2 * N.pi, 200))
plt.plot(xE, yE, label="Earth")
# Plot Orbits
r = res.final("rStart")
xS = r * N.cos(N.linspace(0, 2 * N.pi, 200))
yS = r * N.sin(N.linspace(0, 2 * N.pi, 200))
plt.plot(xS, yS, label="Low Orbit")
r = res.final("rFinal")
xSE = r * N.cos(N.linspace(0, 2 * N.pi, 200))
ySE = r * N.sin(N.linspace(0, 2 * N.pi, 200))
plt.plot(xSE, ySE, label="High Orbit")
# Plot Satellite trajectory
x_sim = N.hstack((N.hstack((x_phase_1, x_phase_2)), x_phase_3))
y_sim = N.hstack((N.hstack((y_phase_1, y_phase_2)), y_phase_3))
plt.plot(x_sim, y_sim, "-", label="Satellite")
# Plot Rocket Burns
plt.arrow(x_phase_1[-1],
y_phase_1[-1],
0.5 * dx1,
0.5 * dy1,
width=0.01,
label="dv1")
plt.arrow(x_phase_2[-1],
y_phase_2[-1],
0.5 * dx2,
0.5 * dy2,
width=0.01,
label="dv2")
plt.legend()
plt.show()
def run_demo(with_plots=True):
"""
This example demonstrates how to solve parameter estimation problmes.
The data used in the example was recorded by Kristian Soltesz at the
Department of Automatic Control.
"""
curr_dir = os.path.dirname(os.path.abspath(__file__))
# Load measurement data from file
data = loadmat(curr_dir + '/files/qt_par_est_data.mat', appendmat=False)
# Extract data series
t_meas = data['t'][6000::100, 0] - 60
y1_meas = data['y1_f'][6000::100, 0] / 100
y2_meas = data['y2_f'][6000::100, 0] / 100
y3_meas = data['y3_d'][6000::100, 0] / 100
y4_meas = data['y4_d'][6000::100, 0] / 100
u1 = data['u1_d'][6000::100, 0]
u2 = data['u2_d'][6000::100, 0]
# Plot measurements and inputs
if with_plots:
plt.figure(1)
plt.clf()
plt.subplot(2, 2, 1)
plt.plot(t_meas, y3_meas)
plt.title('x3')
plt.grid()
plt.subplot(2, 2, 2)
plt.plot(t_meas, y4_meas)
plt.title('x4')
plt.grid()
plt.subplot(2, 2, 3)
plt.plot(t_meas, y1_meas)
plt.title('x1')
plt.xlabel('t[s]')
plt.grid()
plt.subplot(2, 2, 4)
plt.plot(t_meas, y2_meas)
plt.title('x2')
plt.xlabel('t[s]')
plt.grid()
plt.figure(2)
plt.clf()
plt.subplot(2, 1, 1)
plt.plot(t_meas, u1)
plt.hold(True)
plt.title('u1')
plt.grid()
plt.subplot(2, 1, 2)
plt.plot(t_meas, u2)
plt.title('u2')
plt.xlabel('t[s]')
plt.hold(True)
plt.grid()
# Build input trajectory matrix for use in simulation
u = N.transpose(N.vstack((t_meas, u1, u2)))
# compile FMU
fmu_name = compile_fmu('QuadTankPack.Sim_QuadTank',
curr_dir + '/files/QuadTankPack.mop')
# Load model
model = load_fmu(fmu_name)
# Simulate model response with nominal parameters
res = model.simulate(input=(['u1', 'u2'], u), start_time=0., final_time=60)
# Load simulation result
x1_sim = res['qt.x1']
x2_sim = res['qt.x2']
x3_sim = res['qt.x3']
x4_sim = res['qt.x4']
t_sim = res['time']
u1_sim = res['u1']
u2_sim = res['u2']
assert N.abs(res.final('qt.x1') - 0.05642485) < 1e-3
assert N.abs(res.final('qt.x2') - 0.05510478) < 1e-3
assert N.abs(res.final('qt.x3') - 0.02736532) < 1e-3
assert N.abs(res.final('qt.x4') - 0.02789808) < 1e-3
assert N.abs(res.final('u1') - 6.0) < 1e-3
assert N.abs(res.final('u2') - 5.0) < 1e-3
# Plot simulation result
if with_plots:
plt.figure(1)
plt.subplot(2, 2, 1)
plt.plot(t_sim, x3_sim)
plt.subplot(2, 2, 2)
plt.plot(t_sim, x4_sim)
plt.subplot(2, 2, 3)
plt.plot(t_sim, x1_sim)
plt.subplot(2, 2, 4)
plt.plot(t_sim, x2_sim)
plt.figure(2)
plt.subplot(2, 1, 1)
plt.plot(t_sim, u1_sim, 'r')
plt.subplot(2, 1, 2)
plt.plot(t_sim, u2_sim, 'r')
# Compile the Optimica model to an XML file
model_name = compile_fmux("QuadTankPack.QuadTank_ParEstCasADi",
curr_dir + "/files/QuadTankPack.mop")
# Load the model
model_casadi = CasadiModel(model_name)
"""
The collocation algorithm minimizes, if the parameter_estimation_data
option is set, a quadrature approximation of the integral
\int_{t_0}^{t_f} (y(t)-y^{meas}(t))^T Q (y(t)-y^{meas}(t)) dt
The measurement data is given as a matrix where the first
column is time and the following column contains data corresponding
to the variable names given in the measured_variables list.
Notice that input trajectories used in identification experiments
are handled using the errors in variables method, i.e., deviations
from the measured inputs are penalized in the cost function, rather
than forcing the inputs to follow the measurement profile exactly.
The weighting matrix Q may be used to express that inputs are typically
more reliable than than measured outputs.
"""
# Create measurement data
Q = N.diag([1., 1., 10., 10.])
data_x1 = N.vstack([t_meas, y1_meas])
data_x2 = N.vstack([t_meas, y2_meas])
data_u1 = N.vstack([t_meas, u1])
data_u2 = N.vstack([t_meas, u2])
unconstrained = OrderedDict()
unconstrained['qt.x1'] = data_x1
unconstrained['qt.x2'] = data_x2
unconstrained['u1'] = data_u1
unconstrained['u2'] = data_u2
measurement_data = MeasurementData(Q=Q, unconstrained=unconstrained)
opts = model_casadi.optimize_options()
opts['n_e'] = 60
opts['measurement_data'] = measurement_data
res = model_casadi.optimize(algorithm="LocalDAECollocationAlg",
options=opts)
# Load state profiles
x1_opt = res["qt.x1"]
x2_opt = res["qt.x2"]
x3_opt = res["qt.x3"]
x4_opt = res["qt.x4"]
u1_opt = res["qt.u1"]
u2_opt = res["qt.u2"]
t_opt = res["time"]
# Extract optimal values of parameters
a1_opt = res.final("qt.a1")
a2_opt = res.final("qt.a2")
# Print and assert optimal parameter values
print('a1: ' + str(a1_opt * 1e4) + 'cm^2')
print('a2: ' + str(a2_opt * 1e4) + 'cm^2')
a_ref = [0.02656702, 0.02713898]
N.testing.assert_allclose(1e4 * N.array([a1_opt, a2_opt]),
[0.02656702, 0.02713898],
rtol=1e-4)
# Plot
if with_plots:
plt.figure(1)
plt.subplot(2, 2, 1)
plt.plot(t_opt, x3_opt, 'k')
plt.subplot(2, 2, 2)
plt.plot(t_opt, x4_opt, 'k')
plt.subplot(2, 2, 3)
plt.plot(t_opt, x1_opt, 'k')
plt.subplot(2, 2, 4)
plt.plot(t_opt, x2_opt, 'k')
plt.figure(2)
plt.subplot(2, 1, 1)
plt.plot(t_opt, u1_opt, 'k')
plt.subplot(2, 1, 2)
plt.plot(t_opt, u2_opt, 'k')
plt.show()
def run_demo(with_plots=True):
"""
This example is based on a combined cycle power plant (CCPP). The model has
9 states, 128 algebraic variables and 1 control variable. The task is to
minimize the time required to perform a warm start-up of the power plant.
This problem has become highly industrially relevant during the last few
years, due to an increasing need to improve power generation flexibility.
The example consists of the following steps:
1. Simulating the system using a simple input trajectory.
2. Solving the optimal control problem using the result from the first
step to initialize the non-linear program.
3. Verifying the result from the second step by simulating the system
once more usng the optimized input trajectory.
The model was developed by Francesco Casella and was published in
@InProceedings{CFA2011,
author = "Casella, Francesco and Donida, Filippo and {\AA}kesson, Johan",
title = "Object-Oriented Modeling and Optimal Control: A Case Study in
Power Plant Start-Up",
booktitle = "18th IFAC World Congress",
address = "Milano, Italy",
year = 2011,
month = aug
}
"""
### 1. Compute initial guess trajectories by means of simulation
# Locate the Modelica and Optimica code
file_paths = (os.path.join(get_files_path(), "CombinedCycle.mo"),
os.path.join(get_files_path(), "CombinedCycleStartup.mop"))
# Compile the optimization initialization model
init_sim_fmu = compile_fmu("CombinedCycleStartup.Startup6Reference",
file_paths)
# Load the model
init_sim_model = load_fmu(init_sim_fmu)
# Simulate
init_res = init_sim_model.simulate(start_time=0., final_time=10000.)
# Extract variable profiles
init_sim_plant_p = init_res['plant.p']
init_sim_plant_sigma = init_res['plant.sigma']
init_sim_plant_load = init_res['plant.load']
init_sim_time = init_res['time']
# Plot the initial guess trajectories
if with_plots:
plt.close(1)
plt.figure(1)
plt.subplot(3, 1, 1)
plt.plot(init_sim_time, init_sim_plant_p * 1e-6)
plt.ylabel('evaporator pressure [MPa]')
plt.grid(True)
plt.title('Initial guess obtained by simulation')
plt.subplot(3, 1, 2)
plt.plot(init_sim_time, init_sim_plant_sigma * 1e-6)
plt.grid(True)
plt.ylabel('turbine thermal stress [MPa]')
plt.subplot(3, 1, 3)
plt.plot(init_sim_time, init_sim_plant_load)
plt.grid(True)
plt.ylabel('input load [1]')
plt.xlabel('time [s]')
### 2. Solve the optimal control problem
# Compile model
fmux = compile_fmux("CombinedCycleStartup.Startup6", file_paths)
# Load model
opt_model = CasadiModel(fmux)
# Set options
opt_opts = opt_model.optimize_options()
opt_opts['n_e'] = 50 # Number of elements
opt_opts['init_traj'] = init_res.result_data # Simulation result
opt_opts['nominal_traj'] = init_res.result_data
# Solve the optimal control problem
opt_res = opt_model.optimize(options=opt_opts)
# Extract variable profiles
opt_plant_p = opt_res['plant.p']
opt_plant_sigma = opt_res['plant.sigma']
opt_plant_load = opt_res['plant.load']
opt_time = opt_res['time']
opt_input = N.vstack([opt_time, opt_plant_load]).T
# Plot the optimized trajectories
if with_plots:
plt.close(2)
plt.figure(2)
plt.subplot(3, 1, 1)
plt.plot(opt_time, opt_plant_p * 1e-6)
plt.ylabel('evaporator pressure [MPa]')
plt.grid(True)
plt.title('Optimized trajectories')
plt.subplot(3, 1, 2)
plt.plot(opt_time, opt_plant_sigma * 1e-6)
plt.grid(True)
plt.ylabel('turbine thermal stress [MPa]')
plt.subplot(3, 1, 3)
plt.plot(opt_time, opt_plant_load)
plt.grid(True)
plt.ylabel('input load [1]')
plt.xlabel('time [s]')
# Verify solution for testing purposes
try:
import casadi
except:
pass
else:
cost = float(opt_res.solver.solver.output(casadi.NLP_SOLVER_F))
N.testing.assert_allclose(cost, 17492.465548193624, rtol=1e-5)
### 3. Simulate to verify the optimal solution
# Compile model
sim_fmu = compile_fmu("CombinedCycle.Optimization.Plants.CC0D_WarmStartUp",
file_paths)
# Load model
sim_model = load_fmu(sim_fmu)
# Simulate using optimized input
sim_res = sim_model.simulate(start_time=0., final_time=4000.,
input=('load', opt_input))
# Extract variable profiles
sim_plant_p = sim_res['p']
sim_plant_sigma = sim_res['sigma']
sim_plant_load = sim_res['load']
sim_time = sim_res['time']
# Plot the simulated trajectories
if with_plots:
plt.close(3)
plt.figure(3)
plt.subplot(3, 1, 1)
plt.plot(opt_time, opt_plant_p * 1e-6, '--', lw=5)
plt.hold(True)
plt.plot(sim_time, sim_plant_p * 1e-6, lw=2)
plt.ylabel('evaporator pressure [MPa]')
plt.grid(True)
plt.legend(('optimized', 'simulated'), loc='lower right')
plt.title('Verification')
plt.subplot(3, 1, 2)
plt.plot(opt_time, opt_plant_sigma * 1e-6, '--', lw=5)
plt.hold(True)
plt.plot(sim_time, sim_plant_sigma * 1e-6, lw=2)
plt.ylabel('turbine thermal stress [MPa]')
plt.grid(True)
plt.subplot(3, 1, 3)
plt.plot(opt_time, opt_plant_load, '--', lw=5)
plt.hold(True)
plt.plot(sim_time, sim_plant_load, lw=2)
plt.ylabel('input load [1]')
plt.xlabel('time [s]')
plt.grid(True)
plt.show()
# Verify solution for testing purposes
N.testing.assert_allclose(opt_res.final('plant.p'),
sim_res.final('p'), rtol=5e-3)
def run_demo(with_plots=True):
"""
Demonstrate how to solve a simple parameter estimation problem.
"""
# Locate model file
curr_dir = os.path.dirname(os.path.abspath(__file__));
model_path = curr_dir + "/files/ParameterEstimation_1.mop"
# Compile the model into an FMU and FMUX
fmu = compile_fmu("ParEst.SecondOrder", model_path)
fmux = compile_fmux("ParEst.ParEstCasADi", model_path)
# Load the model
sim_model = load_fmu(fmu)
opt_model = CasadiModel(fmux)
# Simulate model with nominal parameters
sim_res = sim_model.simulate(final_time=10)
# Extract nominal trajectories
t_sim = sim_res['time']
x1_sim = sim_res['x1']
x2_sim = sim_res['x2']
# Get measurement data with 1 Hz and add measurement noise
sim_model = load_fmu(fmu)
meas_res = sim_model.simulate(final_time=10, options={'ncp': 10})
x1_meas = meas_res['x1']
noise = [0.01463904, 0.0139424, 0.09834249, 0.0768069, 0.01971631,
-0.03827911, 0.05266659, -0.02608245, 0.05270525, 0.04717024,
0.0779514]
t_meas = N.linspace(0, 10, 11)
y_meas = x1_meas + noise
if with_plots:
# Plot simulation
plt.close(1)
plt.figure(1)
plt.subplot(2, 1, 1)
plt.plot(t_sim, x1_sim)
plt.grid()
plt.plot(t_meas, y_meas, 'x')
plt.ylabel('x1')
plt.subplot(2, 1, 2)
plt.plot(t_sim, x2_sim)
plt.grid()
plt.ylabel('x2')
plt.show()
# Create MeasurementData object
Q = N.array([[1.]])
unconstrained = OrderedDict()
unconstrained['sys.y'] = N.vstack([t_meas, y_meas])
measurement_data = MeasurementData(Q=Q, unconstrained=unconstrained)
# Set optimization options
opts = opt_model.optimize_options()
opts['measurement_data'] = measurement_data
opts['n_e'] = 16
# Optimize
opt_res = opt_model.optimize(options=opts)
# Extract variable profiles
x1_opt = opt_res['sys.x1']
x2_opt = opt_res['sys.x2']
w_opt = opt_res.final('sys.w')
z_opt = opt_res.final('sys.z')
t_opt = opt_res['time']
# Assert results
assert N.abs(opt_res.final('sys.x1') - 1.) < 1e-2
assert N.abs(w_opt - 1.05) < 1e-2
assert N.abs(z_opt - 0.47) < 1e-2
# Plot optimization result
if with_plots:
plt.close(2)
plt.figure(2)
plt.subplot(2, 1, 1)
plt.plot(t_opt, x1_opt, 'g')
plt.plot(t_sim, x1_sim)
plt.plot(t_meas, y_meas, 'x')
plt.grid()
plt.subplot(2, 1, 2)
plt.plot(t_opt, x2_opt, 'g')
plt.grid()
plt.plot(t_sim, x2_sim)
plt.show()
print("** Optimal parameter values: **")
print("w = %f" % w_opt)
print("z = %f" % z_opt)
def run_demo(with_plots=True):
"""
This example is based on a combined cycle power plant (CCPP). The model has
9 states, 128 algebraic variables and 1 control variable. The task is to
minimize the time required to perform a warm start-up of the power plant.
This problem has become highly industrially relevant during the last few
years, due to an increasing need to improve power generation flexibility.
The example consists of the following steps:
1. Simulating the system using a simple input trajectory.
2. Solving the optimal control problem using the result from the first
step to initialize the non-linear program.
3. Verifying the result from the second step by simulating the system
once more usng the optimized input trajectory.
The model was developed by Francesco Casella and was published in
@InProceedings{CFA2011,
author = "Casella, Francesco and Donida, Filippo and {\AA}kesson, Johan",
title = "Object-Oriented Modeling and Optimal Control: A Case Study in
Power Plant Start-Up",
booktitle = "18th IFAC World Congress",
address = "Milano, Italy",
year = 2011,
month = aug
}
"""
### 1. Compute initial guess trajectories by means of simulation
# Locate the Modelica and Optimica code
file_paths = (os.path.join(get_files_path(), "CombinedCycle.mo"),
os.path.join(get_files_path(), "CombinedCycleStartup.mop"))
# Compile the optimization initialization model
init_sim_fmu = compile_fmu("CombinedCycleStartup.Startup6Reference",
file_paths)
# Load the model
init_sim_model = load_fmu(init_sim_fmu)
# Simulate
init_res = init_sim_model.simulate(start_time=0., final_time=10000.)
# Extract variable profiles
init_sim_plant_p = init_res['plant.p']
init_sim_plant_sigma = init_res['plant.sigma']
init_sim_plant_load = init_res['plant.load']
init_sim_time = init_res['time']
# Plot the initial guess trajectories
if with_plots:
plt.close(1)
plt.figure(1)
plt.subplot(3, 1, 1)
plt.plot(init_sim_time, init_sim_plant_p * 1e-6)
plt.ylabel('evaporator pressure [MPa]')
plt.grid(True)
plt.title('Initial guess obtained by simulation')
plt.subplot(3, 1, 2)
plt.plot(init_sim_time, init_sim_plant_sigma * 1e-6)
plt.grid(True)
plt.ylabel('turbine thermal stress [MPa]')
plt.subplot(3, 1, 3)
plt.plot(init_sim_time, init_sim_plant_load)
plt.grid(True)
plt.ylabel('input load [1]')
plt.xlabel('time [s]')
### 2. Solve the optimal control problem
# Compile model
fmux = compile_fmux("CombinedCycleStartup.Startup6", file_paths)
# Load model
opt_model = CasadiModel(fmux)
# Set options
opt_opts = opt_model.optimize_options()
opt_opts['n_e'] = 50 # Number of elements
opt_opts['init_traj'] = init_res.result_data # Simulation result
opt_opts['nominal_traj'] = init_res.result_data
# Solve the optimal control problem
opt_res = opt_model.optimize(options=opt_opts)
# Extract variable profiles
opt_plant_p = opt_res['plant.p']
opt_plant_sigma = opt_res['plant.sigma']
opt_plant_load = opt_res['plant.load']
opt_time = opt_res['time']
opt_input = N.vstack([opt_time, opt_plant_load]).T
# Plot the optimized trajectories
if with_plots:
plt.close(2)
plt.figure(2)
plt.subplot(3, 1, 1)
plt.plot(opt_time, opt_plant_p * 1e-6)
plt.ylabel('evaporator pressure [MPa]')
plt.grid(True)
plt.title('Optimized trajectories')
plt.subplot(3, 1, 2)
plt.plot(opt_time, opt_plant_sigma * 1e-6)
plt.grid(True)
plt.ylabel('turbine thermal stress [MPa]')
plt.subplot(3, 1, 3)
plt.plot(opt_time, opt_plant_load)
plt.grid(True)
plt.ylabel('input load [1]')
plt.xlabel('time [s]')
# Verify solution for testing purposes
try:
import casadi
except:
pass
else:
cost = float(opt_res.solver.solver.output(casadi.NLP_SOLVER_F))
N.testing.assert_allclose(cost, 17492.465548193624, rtol=1e-5)
### 3. Simulate to verify the optimal solution
# Compile model
sim_fmu = compile_fmu("CombinedCycle.Optimization.Plants.CC0D_WarmStartUp",
file_paths)
# Load model
sim_model = load_fmu(sim_fmu)
# Simulate using optimized input
sim_res = sim_model.simulate(start_time=0.,
final_time=4000.,
input=('load', opt_input))
# Extract variable profiles
sim_plant_p = sim_res['p']
sim_plant_sigma = sim_res['sigma']
sim_plant_load = sim_res['load']
sim_time = sim_res['time']
# Plot the simulated trajectories
if with_plots:
plt.close(3)
plt.figure(3)
plt.subplot(3, 1, 1)
plt.plot(opt_time, opt_plant_p * 1e-6, '--', lw=5)
plt.hold(True)
plt.plot(sim_time, sim_plant_p * 1e-6, lw=2)
plt.ylabel('evaporator pressure [MPa]')
plt.grid(True)
plt.legend(('optimized', 'simulated'), loc='lower right')
plt.title('Verification')
plt.subplot(3, 1, 2)
plt.plot(opt_time, opt_plant_sigma * 1e-6, '--', lw=5)
plt.hold(True)
plt.plot(sim_time, sim_plant_sigma * 1e-6, lw=2)
plt.ylabel('turbine thermal stress [MPa]')
plt.grid(True)
plt.subplot(3, 1, 3)
plt.plot(opt_time, opt_plant_load, '--', lw=5)
plt.hold(True)
plt.plot(sim_time, sim_plant_load, lw=2)
plt.ylabel('input load [1]')
plt.xlabel('time [s]')
plt.grid(True)
plt.show()
# Verify solution for testing purposes
N.testing.assert_allclose(opt_res.final('plant.p'),
sim_res.final('p'),
rtol=5e-3)
def run_demo(with_plots=True):
"""
This example demonstrates how to solve parameter estimation problmes.
The data used in the example was recorded by Kristian Soltesz at the
Department of Automatic Control.
"""
curr_dir = os.path.dirname(os.path.abspath(__file__));
# Load measurement data from file
data = loadmat(curr_dir+'/files/qt_par_est_data.mat',appendmat=False)
# Extract data series
t_meas = data['t'][6000::100,0]-60
y1_meas = data['y1_f'][6000::100,0]/100
y2_meas = data['y2_f'][6000::100,0]/100
y3_meas = data['y3_d'][6000::100,0]/100
y4_meas = data['y4_d'][6000::100,0]/100
u1 = data['u1_d'][6000::100,0]
u2 = data['u2_d'][6000::100,0]
# Plot measurements and inputs
if with_plots:
plt.figure(1)
plt.clf()
plt.subplot(2,2,1)
plt.plot(t_meas,y3_meas)
plt.title('x3')
plt.grid()
plt.subplot(2,2,2)
plt.plot(t_meas,y4_meas)
plt.title('x4')
plt.grid()
plt.subplot(2,2,3)
plt.plot(t_meas,y1_meas)
plt.title('x1')
plt.xlabel('t[s]')
plt.grid()
plt.subplot(2,2,4)
plt.plot(t_meas,y2_meas)
plt.title('x2')
plt.xlabel('t[s]')
plt.grid()
plt.figure(2)
plt.clf()
plt.subplot(2,1,1)
plt.plot(t_meas,u1)
plt.hold(True)
plt.title('u1')
plt.grid()
plt.subplot(2,1,2)
plt.plot(t_meas,u2)
plt.title('u2')
plt.xlabel('t[s]')
plt.hold(True)
plt.grid()
# Build input trajectory matrix for use in simulation
u = N.transpose(N.vstack((t_meas,u1,u2)))
# compile FMU
fmu_name = compile_fmu('QuadTankPack.Sim_QuadTank',
curr_dir+'/files/QuadTankPack.mop')
# Load model
model = load_fmu(fmu_name)
# Simulate model response with nominal parameters
res = model.simulate(input=(['u1','u2'],u),start_time=0.,final_time=60)
# Load simulation result
x1_sim = res['qt.x1']
x2_sim = res['qt.x2']
x3_sim = res['qt.x3']
x4_sim = res['qt.x4']
t_sim = res['time']
u1_sim = res['u1']
u2_sim = res['u2']
assert N.abs(res.final('qt.x1') - 0.05642485) < 1e-3
assert N.abs(res.final('qt.x2') - 0.05510478) < 1e-3
assert N.abs(res.final('qt.x3') - 0.02736532) < 1e-3
assert N.abs(res.final('qt.x4') - 0.02789808) < 1e-3
assert N.abs(res.final('u1') - 6.0) < 1e-3
assert N.abs(res.final('u2') - 5.0) < 1e-3
# Plot simulation result
if with_plots:
plt.figure(1)
plt.subplot(2,2,1)
plt.plot(t_sim,x3_sim)
plt.subplot(2,2,2)
plt.plot(t_sim,x4_sim)
plt.subplot(2,2,3)
plt.plot(t_sim,x1_sim)
plt.subplot(2,2,4)
plt.plot(t_sim,x2_sim)
plt.figure(2)
plt.subplot(2,1,1)
plt.plot(t_sim,u1_sim,'r')
plt.subplot(2,1,2)
plt.plot(t_sim,u2_sim,'r')
# Compile the Optimica model to an XML file
model_name = compile_fmux("QuadTankPack.QuadTank_ParEstCasADi",
curr_dir+"/files/QuadTankPack.mop")
# Load the model
model_casadi=CasadiModel(model_name)
"""
The collocation algorithm minimizes, if the parameter_estimation_data
option is set, a quadrature approximation of the integral
\int_{t_0}^{t_f} (y(t)-y^{meas}(t))^T Q (y(t)-y^{meas}(t)) dt
The measurement data is given as a matrix where the first
column is time and the following column contains data corresponding
to the variable names given in the measured_variables list.
Notice that input trajectories used in identification experiments
are handled using the errors in variables method, i.e., deviations
from the measured inputs are penalized in the cost function, rather
than forcing the inputs to follow the measurement profile exactly.
The weighting matrix Q may be used to express that inputs are typically
more reliable than than measured outputs.
"""
# Create measurement data
Q = N.diag([1., 1., 10., 10.])
data_x1 = N.vstack([t_meas, y1_meas])
data_x2 = N.vstack([t_meas, y2_meas])
data_u1 = N.vstack([t_meas, u1])
data_u2 = N.vstack([t_meas, u2])
unconstrained = OrderedDict()
unconstrained['qt.x1'] = data_x1
unconstrained['qt.x2'] = data_x2
unconstrained['u1'] = data_u1
unconstrained['u2'] = data_u2
measurement_data = MeasurementData(Q=Q, unconstrained=unconstrained)
opts = model_casadi.optimize_options()
opts['n_e'] = 60
opts['measurement_data'] = measurement_data
res = model_casadi.optimize(options=opts)
# Load state profiles
x1_opt = res["qt.x1"]
x2_opt = res["qt.x2"]
x3_opt = res["qt.x3"]
x4_opt = res["qt.x4"]
u1_opt = res["qt.u1"]
u2_opt = res["qt.u2"]
t_opt = res["time"]
# Extract optimal values of parameters
a1_opt = res.final("qt.a1")
a2_opt = res.final("qt.a2")
# Print and assert optimal parameter values
print('a1: ' + str(a1_opt*1e4) + 'cm^2')
print('a2: ' + str(a2_opt*1e4) + 'cm^2')
a_ref = [0.02656702, 0.02713898]
N.testing.assert_allclose(1e4 * N.array([a1_opt, a2_opt]),
[0.02656702, 0.02713898], rtol=1e-4)
# Plot
if with_plots:
plt.figure(1)
plt.subplot(2,2,1)
plt.plot(t_opt,x3_opt,'k')
plt.subplot(2,2,2)
plt.plot(t_opt,x4_opt,'k')
plt.subplot(2,2,3)
plt.plot(t_opt,x1_opt,'k')
plt.subplot(2,2,4)
plt.plot(t_opt,x2_opt,'k')
plt.figure(2)
plt.subplot(2,1,1)
plt.plot(t_opt,u1_opt,'k')
plt.subplot(2,1,2)
plt.plot(t_opt,u2_opt,'k')
plt.show()
def run_demo(with_plots=True):
"""
This example is based on the Hicks-Ray Continuously Stirred Tank Reactors
(CSTR) system. The system has two states, the concentration and the
temperature. The control input to the system is the temperature of the
cooling flow in the reactor jacket. The chemical reaction in the reactor is
exothermic, and also temperature dependent; high temperature results in high
reaction rate.
The problem is solved using the CasADi-based collocation algorithm. The
steps performed correspond to those demonstrated in
example pyjmi.examples.cstr, where the same problem is solved using the
default JMI algorithm. FMI is used for initialization and simulation
purposes.
The following steps are demonstrated in this example:
1. How to solve the initialization problem. The initialization model has
equations specifying that all derivatives should be identically zero,
which implies that a stationary solution is obtained. Two stationary
points, corresponding to different inputs, are computed. We call the
stationary points A and B respectively. Point A corresponds to
operating conditions where the reactor is cold and the reaction rate is
low, whereas point B corresponds to a higher temperature where the
reaction rate is high.
2. How to generate an initial guess for a direct collocation method by
means of simulation with a constant input. The trajectories resulting
from the simulation are used to initialize the variables in the
transcribed NLP.
3. An optimal control problem is solved where the objective is to transfer
the state of the system from stationary point A to point B. The
challenge is to ignite the reactor while avoiding uncontrolled
temperature increase.
4. Finally the system is simulated using the optimal control profile. This
step is important in order to verify that the approximation in the
transcription step is sufficiently accurate.
"""
### 1. Solve the initialization problem
# Locate the Modelica and Optimica code
file_path = os.path.join(get_files_path(), "CSTR.mop")
# Compile the stationary initialization model into a FMU
init_fmu = compile_fmu("CSTR.CSTR_Init", file_path)
# Load the FMU
init_model = load_fmu(init_fmu)
# Set input for Stationary point A
Tc_0_A = 250
init_model.set('Tc', Tc_0_A)
# Solve the initialization problem using FMI
init_model.initialize()
# Store stationary point A
[c_0_A, T_0_A] = init_model.get(['c', 'T'])
# Print some data for stationary point A
print(' *** Stationary point A ***')
print('Tc = %f' % Tc_0_A)
print('c = %f' % c_0_A)
print('T = %f' % T_0_A)
# Set inputs for Stationary point B
init_model = load_fmu(init_fmu)
Tc_0_B = 280
init_model.set('Tc', Tc_0_B)
# Solve the initialization problem using FMI
init_model.initialize()
# Store stationary point B
[c_0_B, T_0_B] = init_model.get(['c', 'T'])
# Print some data for stationary point B
print(' *** Stationary point B ***')
print('Tc = %f' % Tc_0_B)
print('c = %f' % c_0_B)
print('T = %f' % T_0_B)
### 2. Compute initial guess trajectories by means of simulation
# Compile the optimization initialization model
init_sim_fmu = compile_fmu("CSTR.CSTR_Init_Optimization", file_path)
# Load the model
init_sim_model = load_fmu(init_sim_fmu)
# Set initial and reference values
init_sim_model.set('cstr.c_init', c_0_A)
init_sim_model.set('cstr.T_init', T_0_A)
init_sim_model.set('c_ref', c_0_B)
init_sim_model.set('T_ref', T_0_B)
init_sim_model.set('Tc_ref', Tc_0_B)
# Simulate with constant input Tc
init_res = init_sim_model.simulate(start_time=0., final_time=150.)
# Extract variable profiles
c_init_sim = init_res['cstr.c']
T_init_sim = init_res['cstr.T']
Tc_init_sim = init_res['cstr.Tc']
t_init_sim = init_res['time']
# Plot the initial guess trajectories
if with_plots:
plt.close(1)
plt.figure(1)
plt.hold(True)
plt.subplot(3, 1, 1)
plt.plot(t_init_sim, c_init_sim)
plt.grid()
plt.ylabel('Concentration')
plt.title('Initial guess obtained by simulation')
plt.subplot(3, 1, 2)
plt.plot(t_init_sim, T_init_sim)
plt.grid()
plt.ylabel('Temperature')
plt.subplot(3, 1, 3)
plt.plot(t_init_sim, Tc_init_sim)
plt.grid()
plt.ylabel('Cooling temperature')
plt.xlabel('time')
plt.show()
### 3. Solve the optimal control problem
# Compile model
fmux = compile_fmux("CSTR.CSTR_Opt2", file_path)
# Load model
cstr = CasadiModel(fmux)
# Set reference values
cstr.set('Tc_ref', Tc_0_B)
cstr.set('c_ref', c_0_B)
cstr.set('T_ref', T_0_B)
# Set initial values
cstr.set('cstr.c_init', c_0_A)
cstr.set('cstr.T_init', T_0_A)
# Set options
opt_opts = cstr.optimize_options()
opt_opts['n_e'] = 100 # Number of elements
opt_opts['init_traj'] = init_res.result_data
opt_opts['nominal_traj'] = init_res.result_data
opt_opts['IPOPT_options']['tol'] = 1e-10
# Solve the optimal control problem
res = cstr.optimize(options=opt_opts)
# Extract variable profiles
c_res = res['cstr.c']
T_res = res['cstr.T']
Tc_res = res['cstr.Tc']
time_res = res['time']
c_ref = res['c_ref']
T_ref = res['T_ref']
Tc_ref = res['Tc_ref']
# Verify solution for testing purposes
try:
import casadi
except:
pass
else:
cost = float(res.solver.solver.output(casadi.NLP_SOLVER_F))
assert(N.abs(cost/1.e3 - 1.8585429) < 1e-3)
# Plot the results
if with_plots:
plt.close(2)
plt.figure(2)
plt.hold(True)
plt.subplot(3, 1, 1)
plt.plot(time_res, c_res)
plt.plot([time_res[0], time_res[-1]], [c_ref, c_ref], '--')
plt.grid()
plt.ylabel('Concentration')
plt.title('Optimized trajectories')
plt.subplot(312)
plt.plot(time_res,T_res)
plt.plot([time_res[0],time_res[-1]],[T_ref,T_ref],'--')
plt.grid()
plt.ylabel('Temperature')
plt.subplot(313)
plt.plot(time_res,Tc_res)
plt.plot([time_res[0],time_res[-1]],[Tc_ref,Tc_ref],'--')
plt.grid()
plt.ylabel('Cooling temperature')
plt.xlabel('time')
plt.show()
### 4. Simulate to verify the optimal solution
# Compile model
sim_fmu = compile_fmu("CSTR.CSTR", file_path)
# Load model
sim_model = load_fmu(sim_fmu)
# Get optimized input
(_, opt_input) = res.solver.get_opt_input()
# Set initial values
sim_model.set('c_init',c_0_A)
sim_model.set('T_init',T_0_A)
# Simulate using optimized input
sim_opts = sim_model.simulate_options()
sim_opts['CVode_options']['rtol'] = 1e-6
sim_opts['CVode_options']['atol'] = 1e-8
res = sim_model.simulate(start_time=0., final_time=150.,
input=('Tc', opt_input), options=sim_opts)
# Extract variable profiles
c_sim=res['c']
T_sim=res['T']
Tc_sim=res['Tc']
time_sim = res['time']
# Verify results
N.testing.assert_array_less(abs(c_res[-1] - c_sim[-1])/c_res[-1], 5e-2)
# Plot the results
if with_plots:
plt.close(3)
plt.figure(3)
plt.hold(True)
plt.subplot(3, 1, 1)
plt.plot(time_res, c_res, '--', lw=5)
plt.plot(time_sim, c_sim, lw=2)
plt.legend(('optimized', 'simulated'))
plt.grid(True)
plt.ylabel('Concentration')
plt.title('Verification')
plt.subplot(3, 1, 2)
plt.plot(time_res, T_res, '--', lw=5)
plt.plot(time_sim, T_sim, lw=2)
plt.grid(True)
plt.ylabel('Temperature')
plt.subplot(3, 1, 3)
plt.plot(time_res, Tc_res, '--', lw=5)
plt.plot(time_sim, Tc_sim, lw=2)
plt.grid(True)
plt.ylabel('Cooling temperature')
plt.xlabel('time')
plt.show()